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The Unreasonable Effectiveness of Mathematics in the Natural Sciences
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is the title of an article published in 1960 by the physicist Eugene Wigner. In the paper, Wigner observed that the mathematical structure of a physical theory often points the way to further advances in that theory and even to empirical predictions.
16 pages, Unknown Binding
First published May 11, 1959
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November 10, 2020
The Grapes of Math
The Unreasonable Effectiveness of Mathematics is a brief but classic essay and fascinating epistemological inquiry written in 1960 by the Nobel prize-winning physicist and mathematician Eugene Wigner.
In it, Wigner considers one of the most perplexing riddles in modern epistemology (the branch of philosophy which examines the nature and structure of knowledge—of how, why, and if we know the things we know, or think we know). The subject of Wigner’s investigation is pretty well summed-up in that pithy title: the unreasonable effectiveness of mathematics in the natural sciences. (It just rolls right off the tongue, doesn’t it?)
To put it in slightly different terms: how is mathematics—a purely deductive, non-empirical pursuit, and one which is often far removed from the messiness of the real world—able to predict with such incredible accuracy the experiential, empirical world around us?
Wigner offers as his first example none other than the cornerstone of our modern scientific mythos: Newton’s discovery of the universal law of gravitation (sans the apocryphal apple, mercifully).
At the time of its conception, Newton’s law—and, more importantly, the mathematical architecture which undergirds it—was only very tenuously tied to any real-world information or empirical data. In fact, the only sound empirical guidance Newton had in taking his fantastic leap was the experimental findings of Galileo, who had been studying the parabolic trajectories of projectiles and uniform acceleration of freely falling bodies, both of which Newton presciently—but prematurely, in purely empirical terms—linked to the mathematically corresponding elliptical shapes described by the orbits of the planets in their heavenly wanderings (which had only relatively recently been given formal mathematical grounding by Johannes Kepler earlier in the 17th century with his three laws of planetary motion; yet another example of math’s “unusual effectiveness,” but one that I’ll forego delving into).
It is, of course, part of Newton’s unique genius that he was able to make such an enormously imaginative leap: to recognize a deeper connection between the parabolic arcs of earthly stones and cannonballs and the celestial ellipses traced out by the planets in their perpetual courses round the sun—and finally, to synthesize it all within a highly generalized mathematical framework.
And all based on that initial imaginative leap.
And yet somehow, some way, this deductive leap, this revolutionary generalization of the behaviors of all massive objects under a universal and mathematically invariant law of gravity, ended up working—and working with absolutely astonishing accuracy.
The full extent of its precision wouldn’t be fully established experimentally until long after Newton’s death—well into the modern era, in fact—and this is further testament to the peculiar enigma of a priori mathematical reasoning, whose true power and accuracy is often only fully realized through empirical means long after the initial mathematical proposition has been fully formulated.
So—whence does this awesome predictive power of seemingly mundane numerical systems, devised by human minds after all, come? Surely mathematics is the true language of nature, linked in countless subtle ways to the most fundamental operations of the natural world, and in a manner far more profound than we can yet fathom.
But, too—why ought there be laws at all—generalizable in universal terms, conveniently equipped as they are with non-local, non-temporal symmetry and invariance? Why don’t the laws of gravity change monthly, like the phases of the moon; or the laws of electromagnetism daily, or hourly, or, for that matter, with unpredictable nonperiodicity—first changing every three months, and then every 1.61 weeks, and then every 0.1123 seconds, and then again every 1,024 years, and so on? Why doesn’t the strong nuclear force behave differently in Brooklyn than it does in Budapest; or act on Earth otherwise than it does on Alpha Centauri?
In other words: why is it that these laws not only exist, but exist in such a way that they are uniquely receptive to mathematical generalization and apprehension by human minds?
How is it that such a broad range of physical phenomena—from the frequency of electromagnetic waves to the movement of particles in a fluid; from the population fluctuations of local fauna to the curve of a conch shell—is describable by a relatively small amount of simple mathematical ratios and constants, like the inverse square law, the Fibonacci sequence, or the ratio of a circle’s circumference to its diameter? (The last of these, pi, pops up unexpectedly in virtually every area of mathematical and scientific inquiry, from engineering to economic modeling to Gaussian distribution in population trends—and therefore, one might understandably be tempted to think that it’s somehow relevant to all of these myriad subjects which, prima facie at least, have nothing to do with circles or geometry at all. Mysteries abound!)
And finally, how is it that mathematics is able to lead us in so many new and unexpected directions with little to no empirical prompting?
It’s much more than the tail wagging the dog—it’s the tail sprouting eyeballs and leading the half-blind dog home in the dark.
Or, to offer a different metaphor, it‘s like making yourself a kite, and then, after years of slowly adding various ornaments and augmentations at random, discovering that those arbitrarily added appendages have somehow transformed your simple kite into an unfathomably powerful machine with the force of a thermonuclear engine, ready for interstellar travel.
~~~
The long history of science contains multitudes of other instances which are illustrative of this wonderfully strange anomaly, this unique power of mathematics, beyond Newton’s insight into the universal nature of gravity. From Pythagoras passing the blacksmith’s forge and listening to the notes different hammers struck as they clanged against the anvil, to Archimedes discovering his fundamental principle of fluid mechanics while reclining in his bathtub, people have long recognized that numbers possess a deep connection to the natural world.
More recent examples, though, illustrate this point in a far more conspicuous and outrageous manner. These range from Maxwell’s electromagnetic field equations yielding up the precise value of the velocity of electromagnetic waves (about 186,000 miles per second), which—as it was the nearly the exact same value as the speed of light, led Maxwell to the eureka realization that light was just one manifestation of electromagnetism, two decades before the existence of electromagnetic waves would be empirically proven by Heinrich Hertz; to Bernhard Riemann devising non-Euclidean geometries for the mathematical modeling of n-dimensional space in the mid-18th century which, though incredibly obscure at the time, would—sixty years later—prove to be the exact mathematical foundation Einstein needed to formulate the equations for his general theory of relativity; to the totally unforeseen prediction by those same Einsteinian equations of a certain kind of cataclysmic stellar gravitational collapse (later dubbed black holes), long before anyone had even imagined such spectacularly strange things could exist in the natural world—least of all Einstein himself!; to Max Planck calculating the Planck length merely as a convenient mathematical workaround, not believing it could have anything more tangible to do with the real world than as a useful mathematical abstraction (as it turned out, of course, it is fundamental to it); to the growing number of subatomic particles that have been predicted by quantum mechanics decades before modes of experimentation sophisticated enough to test their existence are even invented. The list goes on.
For the curious, The Unreasonable Effectiveness of Mathematics is a boldly questing look into an utterly mystifying paradox; a strange crack in the otherwise porcelain-smooth edifice of modern mathematical scientific understanding.*
(*Though it’s true that there are as yet many unresolved disputes in the philosophy of science between adherents of different schools of thought pertaining to various aspects of scientific and mathematical methodology, theory, and understanding—various stripes of logical positivist, antiposivist, critical rationalist, reductionist, instrumentalist, verificationist, inductivist, and so on—guided by the often contradictory conceptualizations of such philosophical colossi as Bertrand Russell, Ludwig Wittgenstein, Karl Popper, and Thomas Kuhn [indeed, Wigner begins his essay with a quote from Russell], these disputes seem, to me at least, to be not so much cracks in the edifice of scientific epistemology as ongoing and necessary renovations to the internal architecture underlying that edifice. Though admittedly, Wigner’s concern, despite being of serious import and interest, may be trivial in the final analysis. After all, if something works so well, why question it? Though alas, questioning things does happen to be the favorite past time of scientist and philosopher alike.)
Wigner, of course, isn’t the only person to have ever spelunked this particular cavern of philosophical controversy and incertitude. But Wigner’s essay is the most influential and easily-digestible, and has inspired a wealth of rejoinders in the sixty years since it was first published, from such later luminaries as quantum physicist Max Tegmark, molecular biologist Arthur Lesk, economist Vela Velupillai, mathematician Ivor Grattan-Guinness, and computer scientist and former director of research at Google, Peter Norvig.
The lasting power and allure of The Unusual Effectiveness of Mathematics lies in the fact that it attempts to excavate and inspect the very foundation stone of mathematical science itself—asking what may very well be an unanswerable question. This makes it frightening. It also makes it fascinating.
LINKS AND FURTHER READING:
*The Unreasonable Effectiveness of Mathematics in the Natural Sciences is available for free as a PDF and is quite short—so if you enjoy underlining or highlighting passages in what you read like I do, you can easily print it out if you want!Here’s a link for download:
https://www.maths.ed.ac.uk/~v1ranick/...
*To read more on Wigner’s paper and the responses it’s provoked over the years, here’s a link to a good overview of his paper, which contains links to the others, on Wikipedia:
https://en.m.wikipedia.org/wiki/The_U...
The Unreasonable Effectiveness of Mathematics is a brief but classic essay and fascinating epistemological inquiry written in 1960 by the Nobel prize-winning physicist and mathematician Eugene Wigner.
In it, Wigner considers one of the most perplexing riddles in modern epistemology (the branch of philosophy which examines the nature and structure of knowledge—of how, why, and if we know the things we know, or think we know). The subject of Wigner’s investigation is pretty well summed-up in that pithy title: the unreasonable effectiveness of mathematics in the natural sciences. (It just rolls right off the tongue, doesn’t it?)
To put it in slightly different terms: how is mathematics—a purely deductive, non-empirical pursuit, and one which is often far removed from the messiness of the real world—able to predict with such incredible accuracy the experiential, empirical world around us?
Wigner offers as his first example none other than the cornerstone of our modern scientific mythos: Newton’s discovery of the universal law of gravitation (sans the apocryphal apple, mercifully).
At the time of its conception, Newton’s law—and, more importantly, the mathematical architecture which undergirds it—was only very tenuously tied to any real-world information or empirical data. In fact, the only sound empirical guidance Newton had in taking his fantastic leap was the experimental findings of Galileo, who had been studying the parabolic trajectories of projectiles and uniform acceleration of freely falling bodies, both of which Newton presciently—but prematurely, in purely empirical terms—linked to the mathematically corresponding elliptical shapes described by the orbits of the planets in their heavenly wanderings (which had only relatively recently been given formal mathematical grounding by Johannes Kepler earlier in the 17th century with his three laws of planetary motion; yet another example of math’s “unusual effectiveness,” but one that I’ll forego delving into).
It is, of course, part of Newton’s unique genius that he was able to make such an enormously imaginative leap: to recognize a deeper connection between the parabolic arcs of earthly stones and cannonballs and the celestial ellipses traced out by the planets in their perpetual courses round the sun—and finally, to synthesize it all within a highly generalized mathematical framework.
And all based on that initial imaginative leap.
And yet somehow, some way, this deductive leap, this revolutionary generalization of the behaviors of all massive objects under a universal and mathematically invariant law of gravity, ended up working—and working with absolutely astonishing accuracy.
The full extent of its precision wouldn’t be fully established experimentally until long after Newton’s death—well into the modern era, in fact—and this is further testament to the peculiar enigma of a priori mathematical reasoning, whose true power and accuracy is often only fully realized through empirical means long after the initial mathematical proposition has been fully formulated.
So—whence does this awesome predictive power of seemingly mundane numerical systems, devised by human minds after all, come? Surely mathematics is the true language of nature, linked in countless subtle ways to the most fundamental operations of the natural world, and in a manner far more profound than we can yet fathom.
But, too—why ought there be laws at all—generalizable in universal terms, conveniently equipped as they are with non-local, non-temporal symmetry and invariance? Why don’t the laws of gravity change monthly, like the phases of the moon; or the laws of electromagnetism daily, or hourly, or, for that matter, with unpredictable nonperiodicity—first changing every three months, and then every 1.61 weeks, and then every 0.1123 seconds, and then again every 1,024 years, and so on? Why doesn’t the strong nuclear force behave differently in Brooklyn than it does in Budapest; or act on Earth otherwise than it does on Alpha Centauri?
In other words: why is it that these laws not only exist, but exist in such a way that they are uniquely receptive to mathematical generalization and apprehension by human minds?
How is it that such a broad range of physical phenomena—from the frequency of electromagnetic waves to the movement of particles in a fluid; from the population fluctuations of local fauna to the curve of a conch shell—is describable by a relatively small amount of simple mathematical ratios and constants, like the inverse square law, the Fibonacci sequence, or the ratio of a circle’s circumference to its diameter? (The last of these, pi, pops up unexpectedly in virtually every area of mathematical and scientific inquiry, from engineering to economic modeling to Gaussian distribution in population trends—and therefore, one might understandably be tempted to think that it’s somehow relevant to all of these myriad subjects which, prima facie at least, have nothing to do with circles or geometry at all. Mysteries abound!)
And finally, how is it that mathematics is able to lead us in so many new and unexpected directions with little to no empirical prompting?
It’s much more than the tail wagging the dog—it’s the tail sprouting eyeballs and leading the half-blind dog home in the dark.
Or, to offer a different metaphor, it‘s like making yourself a kite, and then, after years of slowly adding various ornaments and augmentations at random, discovering that those arbitrarily added appendages have somehow transformed your simple kite into an unfathomably powerful machine with the force of a thermonuclear engine, ready for interstellar travel.
~~~
The long history of science contains multitudes of other instances which are illustrative of this wonderfully strange anomaly, this unique power of mathematics, beyond Newton’s insight into the universal nature of gravity. From Pythagoras passing the blacksmith’s forge and listening to the notes different hammers struck as they clanged against the anvil, to Archimedes discovering his fundamental principle of fluid mechanics while reclining in his bathtub, people have long recognized that numbers possess a deep connection to the natural world.
More recent examples, though, illustrate this point in a far more conspicuous and outrageous manner. These range from Maxwell’s electromagnetic field equations yielding up the precise value of the velocity of electromagnetic waves (about 186,000 miles per second), which—as it was the nearly the exact same value as the speed of light, led Maxwell to the eureka realization that light was just one manifestation of electromagnetism, two decades before the existence of electromagnetic waves would be empirically proven by Heinrich Hertz; to Bernhard Riemann devising non-Euclidean geometries for the mathematical modeling of n-dimensional space in the mid-18th century which, though incredibly obscure at the time, would—sixty years later—prove to be the exact mathematical foundation Einstein needed to formulate the equations for his general theory of relativity; to the totally unforeseen prediction by those same Einsteinian equations of a certain kind of cataclysmic stellar gravitational collapse (later dubbed black holes), long before anyone had even imagined such spectacularly strange things could exist in the natural world—least of all Einstein himself!; to Max Planck calculating the Planck length merely as a convenient mathematical workaround, not believing it could have anything more tangible to do with the real world than as a useful mathematical abstraction (as it turned out, of course, it is fundamental to it); to the growing number of subatomic particles that have been predicted by quantum mechanics decades before modes of experimentation sophisticated enough to test their existence are even invented. The list goes on.
For the curious, The Unreasonable Effectiveness of Mathematics is a boldly questing look into an utterly mystifying paradox; a strange crack in the otherwise porcelain-smooth edifice of modern mathematical scientific understanding.*
(*Though it’s true that there are as yet many unresolved disputes in the philosophy of science between adherents of different schools of thought pertaining to various aspects of scientific and mathematical methodology, theory, and understanding—various stripes of logical positivist, antiposivist, critical rationalist, reductionist, instrumentalist, verificationist, inductivist, and so on—guided by the often contradictory conceptualizations of such philosophical colossi as Bertrand Russell, Ludwig Wittgenstein, Karl Popper, and Thomas Kuhn [indeed, Wigner begins his essay with a quote from Russell], these disputes seem, to me at least, to be not so much cracks in the edifice of scientific epistemology as ongoing and necessary renovations to the internal architecture underlying that edifice. Though admittedly, Wigner’s concern, despite being of serious import and interest, may be trivial in the final analysis. After all, if something works so well, why question it? Though alas, questioning things does happen to be the favorite past time of scientist and philosopher alike.)
Wigner, of course, isn’t the only person to have ever spelunked this particular cavern of philosophical controversy and incertitude. But Wigner’s essay is the most influential and easily-digestible, and has inspired a wealth of rejoinders in the sixty years since it was first published, from such later luminaries as quantum physicist Max Tegmark, molecular biologist Arthur Lesk, economist Vela Velupillai, mathematician Ivor Grattan-Guinness, and computer scientist and former director of research at Google, Peter Norvig.
The lasting power and allure of The Unusual Effectiveness of Mathematics lies in the fact that it attempts to excavate and inspect the very foundation stone of mathematical science itself—asking what may very well be an unanswerable question. This makes it frightening. It also makes it fascinating.
LINKS AND FURTHER READING:
*The Unreasonable Effectiveness of Mathematics in the Natural Sciences is available for free as a PDF and is quite short—so if you enjoy underlining or highlighting passages in what you read like I do, you can easily print it out if you want!Here’s a link for download:
https://www.maths.ed.ac.uk/~v1ranick/...
*To read more on Wigner’s paper and the responses it’s provoked over the years, here’s a link to a good overview of his paper, which contains links to the others, on Wikipedia:
https://en.m.wikipedia.org/wiki/The_U...
November 10, 2019
My Medium post summarizing this classic article: https://medium.com/cantors-paradise/t....
April 26, 2022
This brief essay asks why math proves so effective for describing / codifying physical laws, and whether our physical theories -- built on (phenomenally successful) mathematics -- offer the truth, the whole truth, and nothing but the truth.
There’s a popular story in which a drunk man is found on his hands and knees under a lamppost at night when a police officer comes along. The cops says, “What-cha doin’?” To which the drunk replies, “I dropped my keys, and I’m looking for them?” So, the cop says, “Well, they’re clearly not where you’re looking, why not look elsewhere?” And the drunk says, “Cuz this is where the light is.” I think this story can help us understand what Wigner is getting on about, if only we replace the drunk’s “light” with the scientist’s “elegant mathematics.” Wigner reflects upon why it should be that so many laws of nature seem to be independent from all but a few variables (which is the only way scientists could have discovered them --historically, mathematically, and realistically speaking.) On the other hand, could it be that Physics has led itself into epistemological cul-de-sacs by chasing elegant mathematics?
There’s no doubt that (for whatever the reason turns out to be) mathematics has been tremendously successful in facilitating the construction of theories that make predictions that can be tested with high levels of accuracy. However, that doesn’t mean that some of those theories won’t prove to be mirages.
A few of the examples used in this paper are somewhat esoteric and won’t be readily understood by the average (non-expert) reader. That said, Wigner puts his basic arguments and questions in reasonably clear (if academic) language. The essay is definitely worth reading for its thought-provoking insights.
There’s a popular story in which a drunk man is found on his hands and knees under a lamppost at night when a police officer comes along. The cops says, “What-cha doin’?” To which the drunk replies, “I dropped my keys, and I’m looking for them?” So, the cop says, “Well, they’re clearly not where you’re looking, why not look elsewhere?” And the drunk says, “Cuz this is where the light is.” I think this story can help us understand what Wigner is getting on about, if only we replace the drunk’s “light” with the scientist’s “elegant mathematics.” Wigner reflects upon why it should be that so many laws of nature seem to be independent from all but a few variables (which is the only way scientists could have discovered them --historically, mathematically, and realistically speaking.) On the other hand, could it be that Physics has led itself into epistemological cul-de-sacs by chasing elegant mathematics?
There’s no doubt that (for whatever the reason turns out to be) mathematics has been tremendously successful in facilitating the construction of theories that make predictions that can be tested with high levels of accuracy. However, that doesn’t mean that some of those theories won’t prove to be mirages.
A few of the examples used in this paper are somewhat esoteric and won’t be readily understood by the average (non-expert) reader. That said, Wigner puts his basic arguments and questions in reasonably clear (if academic) language. The essay is definitely worth reading for its thought-provoking insights.
June 12, 2022
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
The question popped into my mind while trolling a New Atheist on his Wordpress blog. What is the relation between mathematics and science?
Math makes use of abstract concepts that don’t exist in the physical world (a perfectly straight line, infinite sets). Science is empirical study of physical phenomena, yet these phenomena can be abstracted away into mathematical concepts. Much of physics, for example, studies phenomena that cannot exist in the material world (frictionless surface, perfect vacuum). Where does the immaterial end and the material begin? Eugene make it a point that this is only by articles of faith, such as invariance (uniformitarianism).
I’m reminded of CS Lewis’ response to Hume on the topic of miracles (p. 204 here).
I only wish that it’d be a requirement for scientists to take more literature courses because wow this was a dry read.
The question popped into my mind while trolling a New Atheist on his Wordpress blog. What is the relation between mathematics and science?
Math makes use of abstract concepts that don’t exist in the physical world (a perfectly straight line, infinite sets). Science is empirical study of physical phenomena, yet these phenomena can be abstracted away into mathematical concepts. Much of physics, for example, studies phenomena that cannot exist in the material world (frictionless surface, perfect vacuum). Where does the immaterial end and the material begin? Eugene make it a point that this is only by articles of faith, such as invariance (uniformitarianism).
I’m reminded of CS Lewis’ response to Hume on the topic of miracles (p. 204 here).
I only wish that it’d be a requirement for scientists to take more literature courses because wow this was a dry read.
February 5, 2024
I was kinda disappointed. He didn’t really make a clear point but the examples were chill.
Read
November 22, 2025"...the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it."
November 24, 2020
vou resumir: escrito por um religioso
October 10, 2020
There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate, The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is PI.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”
October 20, 2021
Fascinating but fairly densely written. I really couldn’t explain this in any depth to someone - will need to track down a summary for people of normal intelligence.
“The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.
“The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.”
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”
“The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.
“The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.”
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”
February 11, 2025
"(...) it is not at all natural that 'laws of nature' exist, much less that man is able to discover them."
"(...) the concepts of mathematics are not chosen for their conceptual simplicity - even sequences of pairs of numbers are far from being the simplest concepts - but for their amenability to clever manipulations and to striking, brilliant arguments"
"(...) the mathematical formulation of the physicist’s often crude experience leads in an uncanny
number of cases to an amazingly accurate description of a large class of phenomena. This shows
that the mathematical language has more to commend it than being the only language which we can
speak; it shows that it is, in a very real sense, the correct language."
It's nice to marvel at how beautifully everything fits together sometimes. We are so lucky that the 'laws of nature' just ignore so many variables that would make them infinitely complex to state, and therefore that the natural sciences are even possible. I enjoy the notion that all physicists do is predicting the future.
"(...) the concepts of mathematics are not chosen for their conceptual simplicity - even sequences of pairs of numbers are far from being the simplest concepts - but for their amenability to clever manipulations and to striking, brilliant arguments"
"(...) the mathematical formulation of the physicist’s often crude experience leads in an uncanny
number of cases to an amazingly accurate description of a large class of phenomena. This shows
that the mathematical language has more to commend it than being the only language which we can
speak; it shows that it is, in a very real sense, the correct language."
It's nice to marvel at how beautifully everything fits together sometimes. We are so lucky that the 'laws of nature' just ignore so many variables that would make them infinitely complex to state, and therefore that the natural sciences are even possible. I enjoy the notion that all physicists do is predicting the future.
December 12, 2023
Wigner delving into the interlinking of mathematics and physics and how it affects our very understanding of everything - every law; every thing we know to be true - really plunged me into a world of speculation. He goes incredibly philosophical in some elements of his arguments and attempts to propose some innovative critiques to some of our beloved laws of nature.
I read this as an ebook here: https://www.maths.ed.ac.uk/~v1ranick/... - this was uploaded by Edinburgh mathematician Andrew Ranicki, who has now unfortunately passed. Nevertheless, if you look at his main page (https://www.maths.ed.ac.uk/~v1ranick/) - he's really uploaded some exceptional stuff... including this book.
Glad I read this; glad I discovered Prof. Ranicki; glad I discovered LAMB SHIFT (how photons could have such an impact on a single hydrogen atom is pretty mind-boggling)!!!
I read this as an ebook here: https://www.maths.ed.ac.uk/~v1ranick/... - this was uploaded by Edinburgh mathematician Andrew Ranicki, who has now unfortunately passed. Nevertheless, if you look at his main page (https://www.maths.ed.ac.uk/~v1ranick/) - he's really uploaded some exceptional stuff... including this book.
Glad I read this; glad I discovered Prof. Ranicki; glad I discovered LAMB SHIFT (how photons could have such an impact on a single hydrogen atom is pretty mind-boggling)!!!
July 21, 2023
blehg
If you believe we are searching for "the ultimate truth" that is to be found in some ultimate reality then Wigner's confusion is inevitable. (math describes a 'mathematical reality' while physics describes 'physical reality' so how do they interact?!?!)
If you believe that mathematical-physics and math are just fields that use similar tools and are devoted to giving objective solutions to problems that we made then this seems like less of a problem.
Old theories should be rejected because they no longer answer the questions we are asking. I don't see how a theory could be measured against reality when that reality is what what the theory is trying to capture.
If you believe we are searching for "the ultimate truth" that is to be found in some ultimate reality then Wigner's confusion is inevitable. (math describes a 'mathematical reality' while physics describes 'physical reality' so how do they interact?!?!)
If you believe that mathematical-physics and math are just fields that use similar tools and are devoted to giving objective solutions to problems that we made then this seems like less of a problem.
Old theories should be rejected because they no longer answer the questions we are asking. I don't see how a theory could be measured against reality when that reality is what what the theory is trying to capture.
May 8, 2020
What I got more out of this paper than the topic itself was the discussion about the purpose of mathematics. We invented advanced mathematical concepts, not for the sake of finding some ultimate truth, but to demonstrate our ingenuity of using these concepts to bring theories that express supreme formal beauty. In this sense, the work of mathematic is a work of art, in which the aesthetic sense doesn't come from the perception of the outside world but the structure of pure logical thoughts in our brain.
October 3, 2024
One of the topics that is the nearest and dearest to my heart and mind, mathematical philosophy and scientific philosophy. On the "article of faith" of theoretical physicists, I think as a Christian it is very easy and natural to understand them. Namely, first, a rational God designed and created this world in a rational way; second, men were created in God's image so they are capable of understanding the law of nature which is described in the mathematical language. First read of scientific philosophy in many years, didn't disappoint.
April 11, 2021
Il libro è ben scritto e anche interessante per certi versi, nonostante la su brevità e poche argomentazioni. Il suo problema fondamentale è che si pone una domanda alla quale non da' alcun tipo di risposta, gira un po' intorno, sonda il terreno e la domanda rimane. Così, la postfazione a fine libro diventa quasi più interessante e accurata del libro stesso. Forse il titolo contribuisce a questo inganno al lettore.
May 22, 2024
Un breve ma stimolante saggio. Perché la matematica funziona così bene per descrivere la natura? Tendiamo a darlo per scontanto, ma è effettivamente una questione su cui riflettere.
Questo libro introduce questa domanda, ma non approfondisce l'argomento. Lo consiglio quindi come primo passo per approcciarsi alla filosofia della matematica. La lettura è scorrevole.
Questo libro introduce questa domanda, ma non approfondisce l'argomento. Lo consiglio quindi come primo passo per approcciarsi alla filosofia della matematica. La lettura è scorrevole.
August 3, 2022
"Their accuracy may not prove their truth and consistency." A very interesting perspective on the application of our abstract thinking to the real world but also pointing out that our thinking is part of the real world.
April 4, 2024
Very interesting and readable article on the role of mathematics in physics and why it's a bloody miracle that mathematics is a useful tool for physics. There's no reason why it should work, it just does.
January 13, 2020
‘The preceding discussion is intended to remind us, first, that it is not at all natural that "laws of nature" exist, much less that man is able to discover them.’
November 12, 2020
Come già prevedevo, non ne ho capito nulla. Interessante per chi ne capisce ovviamente.
March 27, 2022
Un po' inutile, pone la questione del titolo ma il problema non viene affrontato in modo adeguato
July 8, 2022
I have often been puzzled by this question, what is mathematics and what does it have to do with reality, or maybe it is reality?
I really enjoyed the questions posed by the author
I really enjoyed the questions posed by the author
June 14, 2025
No value, no new ideas and not well written
January 17, 2025
Mathematicians patting themselves on the back or a fascinating take on the effectiveness of mathematics?
Sometime way after undergraduate math, it actually becomes interesting and flirts with philosophy that demonstrates great symmetry between branches. Geometry and algebra are dialects of the same language. You can translate between the two.
We all know the constant pi, something to do with the relationship of the diameter of a circle and its circumference, but how the heck is part of the equation that demonstrates the bell curve or distribution of height of people? The same equation predicts that balls will fall at the same rate (not just Italy) and how planets find a path around the sun.
All very interesting.
Sometime way after undergraduate math, it actually becomes interesting and flirts with philosophy that demonstrates great symmetry between branches. Geometry and algebra are dialects of the same language. You can translate between the two.
We all know the constant pi, something to do with the relationship of the diameter of a circle and its circumference, but how the heck is part of the equation that demonstrates the bell curve or distribution of height of people? The same equation predicts that balls will fall at the same rate (not just Italy) and how planets find a path around the sun.
All very interesting.
October 25, 2018
I much preferred Richard Hamming's The Unreasonable Effectiveness of Mathematics.
November 18, 2025
Nonostante sia un libretto molto famoso tra coloro i quali si dedicano alle discipline STEM gli manca qualcosa, imposta il problema ma poi non prova a risolverlo, lettura comunque interessante per riordinare dei pensieri che chiunque abbia studiato discipline tecnico-scientifiche si è ritrovato ad avere
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June 15, 2021Provokes thought. It truly is strange to me how some "coarse" theories (like the Bohr model) can explain a fairly wide range of phenomena. Or how basic mathematical concepts, when applied to physics, can accidentally generalize so well.
February 23, 2015
A hardly guessed bitterness
July 25, 2016
Necessary read to get to the bottom of some mathematics questions that I currently have.
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